Average Error: 53.0 → 0.3
Time: 18.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.026683657222892120941537541511934250593:\\ \;\;\;\;\log \left(\frac{0.125}{x \cdot \left(x \cdot x\right)} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8807071635272687437634431262267753481865:\\ \;\;\;\;\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{\frac{\frac{1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)}{1}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{x \cdot \left(x \cdot x\right)}\right) + x\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.026683657222892120941537541511934250593:\\
\;\;\;\;\log \left(\frac{0.125}{x \cdot \left(x \cdot x\right)} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.8807071635272687437634431262267753481865:\\
\;\;\;\;\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{\frac{\frac{1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)}{1}}{\sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{x \cdot \left(x \cdot x\right)}\right) + x\right)\right)\\

\end{array}
double f(double x) {
        double r10583875 = x;
        double r10583876 = r10583875 * r10583875;
        double r10583877 = 1.0;
        double r10583878 = r10583876 + r10583877;
        double r10583879 = sqrt(r10583878);
        double r10583880 = r10583875 + r10583879;
        double r10583881 = log(r10583880);
        return r10583881;
}
↓
double f(double x) {
        double r10583882 = x;
        double r10583883 = -1.0266836572228921;
        bool r10583884 = r10583882 <= r10583883;
        double r10583885 = 0.125;
        double r10583886 = r10583882 * r10583882;
        double r10583887 = r10583882 * r10583886;
        double r10583888 = r10583885 / r10583887;
        double r10583889 = 0.5;
        double r10583890 = r10583889 / r10583882;
        double r10583891 = 0.0625;
        double r10583892 = 5.0;
        double r10583893 = pow(r10583882, r10583892);
        double r10583894 = r10583891 / r10583893;
        double r10583895 = r10583890 + r10583894;
        double r10583896 = r10583888 - r10583895;
        double r10583897 = log(r10583896);
        double r10583898 = 0.8807071635272687;
        bool r10583899 = r10583882 <= r10583898;
        double r10583900 = 1.0;
        double r10583901 = sqrt(r10583900);
        double r10583902 = r10583882 / r10583901;
        double r10583903 = log(r10583901);
        double r10583904 = r10583902 + r10583903;
        double r10583905 = 0.16666666666666666;
        double r10583906 = r10583905 * r10583887;
        double r10583907 = r10583906 / r10583900;
        double r10583908 = r10583907 / r10583901;
        double r10583909 = r10583904 - r10583908;
        double r10583910 = r10583890 - r10583888;
        double r10583911 = r10583910 + r10583882;
        double r10583912 = r10583882 + r10583911;
        double r10583913 = log(r10583912);
        double r10583914 = r10583899 ? r10583909 : r10583913;
        double r10583915 = r10583884 ? r10583897 : r10583914;
        return r10583915;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0266836572228921

    1. Initial program 62.8

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around -inf 0.3

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{x \cdot \left(x \cdot x\right)} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)}\]

    if -1.0266836572228921 < x < 0.8807071635272687

    1. Initial program 58.6

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]

    3. Simplified0.4

      \[\leadsto \color{blue}{\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{\frac{\frac{1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)}{1}}{\sqrt{1}}}\]

    if 0.8807071635272687 < x

    1. Initial program 32.2

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around inf 0.1

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\]

    3. Simplified0.1

      \[\leadsto \log \left(x + \color{blue}{\left(\left(\frac{0.5}{x} - \frac{0.125}{x \cdot \left(x \cdot x\right)}\right) + x\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.026683657222892120941537541511934250593:\\ \;\;\;\;\log \left(\frac{0.125}{x \cdot \left(x \cdot x\right)} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8807071635272687437634431262267753481865:\\ \;\;\;\;\left(\frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right) - \frac{\frac{\frac{1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)}{1}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{0.5}{x} - \frac{0.125}{x \cdot \left(x \cdot x\right)}\right) + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))